Let $R(\mu \mid \nu ) = \int_{\mathbb R} \log \frac{d\mu}{d\nu} d\nu$ for $\mu, \nu$ probability measures over $\mathbb R$.
By the varational representation formula of Donsker and Varadhan we know that
$$R(\mu \mid \nu ) = \sup_{f \in C_b(\mathbb R)} \left( \int f(x) \mu(dx) - \log \int_R e^{f(x)} \nu(dx) \right).$$
where $C_b$ is the set of continuous bounded functions over $\mathbb R$.
Instead of $C_b$ one could also use all bounded measurable functions.
Now the question:
Is it possible to use also $C_c$ (the set of continuous functions with compact support) in the supremum?
i.e.: Is
$$\sup_{f \in C_b(\mathbb R)} \left( \int f(x) \mu(dx) - \log \int_R e^{f(x)} \nu(dx) \right) = \sup_{f \in C_c(\mathbb R)} \left( \int f(x) \mu(dx) - \log \int_R e^{f(x)} \nu(dx) \right)$$
when $\mu$ and $\nu$ are probability measures?